then \({\bf B}\)  is the so-called generalized spiked sample covariance matrix. Denote \(\lambda _{1},\lambda _{2},\ldots ,\lambda _{M},\lambda _{M+1},\ldots ,\lambda _{p}\)  as the eigenvalues of the \({\bf B}\) . The above model includes the spiked population model as a special case, and the spiked population model was orginally proposed by [1]}. Other extensions on this topic can be found in [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, and so on.
Define the singular value decomposition of \({\bf T}_p\)  as
\({\bf T}_p={\bf V}\left(\begin{array}{cc}{\bf D}_{1}^\frac{1}{2} & 0\\0 & {\bf D}_{2}^\frac{1}{2}\end{array}\right){\bf U}^{\ast }\) 
